In this paper, we focus on barycentric weights and Lebesgue constants for Lagrange interpolation of arbitrary node distributions on \([-1,1]\). The following three main works are included: estimates of upper and lower bounds on the barycentric weights are given in terms of the logarithmic potential function; for interpolation of non-equilibrium potentials, lower bounds with exponentially growing parts of Lebesgue constants are given; and for interpolation consistent with equilibrium potentials, non-exponentially growing upper bounds on their Lebesgue constants are given. Based on the work in this paper, we can discuss the behavior of the Lebesgue constant and the existence of exponential convergence in a unified manner in the framework of potential theory.

翻译：本文重点研究 \([-1,1]\) 区间上任意节点分布的Lagrange插值的重心权值与Lebesgue常数。主要工作包括以下三方面：利用对数势函数给出重心权值上下界的估计；针对非平衡势的插值，给出Lebesgue常数指数增长部分的下界；针对与平衡势一致的插值，给出其Lebesgue常数非指数增长的上界。基于本文工作，可在势理论框架下统一讨论Lebesgue常数的行为以及指数收敛性的存在性。